Question

If \( ax + by = 1 \) is a normal to the parabola \( y^2 = 4px \), then the condition is

• A. \( 4ab = a^2 + b^2 \)
• B. \( 4pab + ab^3 = a^2b^2 \)
• C. \( pa^3 = b^2 - 2pab^2 \)
• D. \( pa^2 + 4pa = a + b \)

Hint:

To find conditions for normals to a parabola, use parametric coordinates and compare forms.

Answer 1

The Correct Option is C

The general equation of a normal to the parabola \( y^2 = 4px \) at point \( (pt^2, 2pt) \) is given by:
\( y = -tx + 2pt + pt^3 \). Rewriting in the form \( ax + by = 1 \), we compare it with the given line.
Through derivation and elimination, the condition becomes:
\[ pa^3 = b^2 - 2pab^2 \]